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On the nontrivial solvability of systems of homogeneous linear equations over ℤ in ZFC

Jan Šaroch — 2020

Commentationes Mathematicae Universitatis Carolinae

Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals κ , an arbitrary nonempty system S of homogeneous ℤ -linear equations is nontrivially solvable in ℤ provided that each of its subsystems of cardinality less than κ is nontrivially solvable in ℤ ?

Bi-ideal-simple semirings

Václav Flaška;  Tomáš Kepka;  Jan Šaroch — 2005

Commentationes Mathematicae Universitatis Carolinae

Commutative congruence-simple semirings were studied in [2] and [7] (but see also [1], [3]--[6]). The non-commutative case almost (see [8]) escaped notice so far. Whatever, every congruence-simple semiring is bi-ideal-simple and the aim of this very short note is to collect several pieces of information on these semirings.

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